This work presents a mechanical reinterpretation of hydrogenic energy levels and the ionization threshold based on the concept of a gradient-stiff vacuum. Building on the framework of extended electron structures, the study demonstrates that the competition between internal gradient pressure and external Coulombic compression breaks the scale invariance of the free state, establishing a unique ground-state length scale. Discrete hydrogenic states emerge as admissible scales under topological constraints, providing a structural rationale for the n² scaling of atomic energy levels. The ionization threshold is reinterpreted as the structural yield strength required to transition between bound and free morphologies of the electron distribution. This perspective offers a geometric and structural foundation for quantization, highlighting the interplay between vacuum stiffness, scale symmetry breaking, and topological constraints. The detailed dynamical coupling between the stationary electron core and the guiding field is left for future investigation, positioning this work as a conceptual bridge toward a fully dynamic “double-solution” framework.
Ping Zhang (Sun,) studied this question.